Free-form bargaining experiment

Mia Lu
Martin Stancsics

Introduction

Motivation

  • Much work on bargaining in experimental economics (understatement)
  • Much less work on free-form bargaining
    • especially between more than two players
  • The one indispensable player / multiple small players setting has real-world relevance
    • Wage bargaining
    • An inventor with an idea and multiple investors
    • A band, where one member owns the PA system

Research question

  • Problem: non-cooperative game theory cannot provide predictions without structure
    • E.g. timing of the game, who makes the offers
    • NCGT solution is alternating offer games, but a lot depends on minor details (Hart and Mas-Colell 1996)
  • How does bargaining power affect bargaining outcomes?
  • How well do cooperative game theory solution concepts describe the outcomes?

What we do

  • Free-form bargaining between three players
    • Almost no structure
    • Group-level unrestricted chat
    • An interface for proposing and accepting allocations
    • No binding decision until the very last second
  • Vary the bargaining power of the indispensable player
    • How important it is to have all small players on board
  • We test, whether:
    • Outcomes vary as we would expect, based on bargaining power
    • Certain CGT solution concepts provide good predictions

Literature

  • Early unstructured bargaining papers (1950s-1990s):
    • E.g. Kalisch et al. (1952), Maschler (1965), Nydegger and Owen (1874), Rapoport and Kahan (1976), Murnighan and Roth (1977), Murnighan and Roth (1978), Michener et al. (1979), Michener and Potter (1981), Leopold-Wildburger (1992)
    • Face to face bargaining, different experimental standards
  • Free-form bargaining
    • E.g. Galeotti, Montero, and Poulsen (2018), Hossain, Lyons, and Siow (2020), Navarro and Veszteg (2020)
    • Almost always bilateral
  • Multi-lateral bargaining
    • E.g. Montero, Sefton, and Zhang (2008), Mitsutsune and Adachi (2014), Tremewan and Vanberg (2016), Chessa et al. (2023), Shinoda and Funaki (2022)
    • Structured or semi-structured
  • Fairness views in bargaining
    • E.g. Luhan, Poulsen, and Roos (2019), Schwaninger (2022), De Clippel and Rozen (2022)

Game and solution concepts

The game

  • Players: \(N = \{A, B_1, B_2\}\)
  • Value function: \(v: 2^N \to \mathbb{R}\)
    • No one can create any value alone: \(v(\{A\}) = v(\{B_i\}) = 0\)
    • Player \(A\) is indispensable: \(v(\{B_1, B_2\}) = 0\)
    • Small players contribute to the value: \(v(\{A, B_i\}) = Y \in [0, 100]\)
    • The more small players the better: \(v(\{A, B_1, B_2\}) = 100\)

How to divide the value between the players?

Shapley value

  • Each players gets their average marginal contribution
    • Fairness based motivation
  • Appealing characterization
    • Efficient: whole pie is distributed
    • Symmetric: identical (from the pov of \(v\)) players should get the same payoff
    • Dummy player property: players contributing nothing get nothing
    • Linearity (across games)
  • Useful properties
    • Always exists
    • Always unique

Core

  • Excess: what a coalition gets vs what it could achieve on its own
  • The set of payoff vectors such that excess is non-negative for all coalitions
    • No coalition has an incentive to deviate
    • Stability-based concept
    • Similar idea to the Nash equilibrium, but for multi-player deviations
  • Very intuitive and plausible, but
    • Can be empty (existence not guaranteed)
    • Can be multi-valued (uniqueness not guaranteed)

Nucleolus

  • Maximizes the smallest excess across coalitions
    • if the core is non-empty, the nucleolus is in the core
    • Mix of stability and fairness
  • Useful properties
    • Always exists
    • Always unique
  • Can be thought of as a way to mix the stability-based intuition behind the core with useful properties of the Shapley value

In our game

2024-10-15T20:21:08.722885 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Experimental design

General structure

  • Conducted in the BLU lab (May 2024)
    • 4 treatments/sessions: \(Y=10\), \(Y=30\), \(Y=90\), dummy player
    • 144 subjects in total (36 subjects per session)
  • Timing:
    • Instructions with comprehension checks
    • Slider task
    • Trial round + 5 bargaining rounds (5 minutes each)
    • Survey (demographics, reasoning, axioms)
  • Average payoff across all rounds + show-up fee
    • Conversion: 1 point = CHF 0.6

Role and group assignment

  • Player roles are assigned stochastically based on players’ performance in the slider task
    • the better they perform, the higher the likelihood of becoming Player A
  • Stranger matching for bargaining groups each round
    • no set of subjects is matched twice
  • Split each session into 6 matching groups (à 6 players) in order to account for dependence between rounds in the analysis
    • Bargaining groups are only redrawn within matching groups

Bargaining

  • Bargaining via public chat and interface for submitting proposals and current acceptances
  • Free-form bargaining:
    • Unlimited number of proposals
    • No restrictions on the order of proposals and acceptances
    • Acceptances are not binding and can be changed any time
  • At the end of each bargaining round:
    • Current acceptances are taken as final decisions
    • A proposal is successful only if all members of a coalition agree on it

Main results

Payoffs – aggregate

2024-10-15T20:21:11.148914 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

  • Dummy player gets something
  • \(Y=10\) and \(Y=30\) treatments look similar
  • Player A gets more in \(Y=30\) treatment
  • Nucleolus better in terms of ‘shape’ but worse in terms of ‘distance’

Payoff of Player A – regression

Model cont. Model ind.
const 32.90*** 34.16***
(0.72) (0.72)
Y 0.08***
(0.02)
Y = 30 0.72
(0.84)
Y = 90 6.49***
(1.70)
N 174 174

Matching-group-level clustered standard errors in parentheses.
* p<.1, ** p<.05, ***p<.01

Payoffs – a deeper look

2024-10-15T20:21:57.956215 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

  • In all treatments, equal(ish) split is a frequent outcome
    • Even in the dummy player treatment!
  • \(Y=10\) and \(Y=30\) treatments still look similar
  • In the \(Y=90\) treatment, A can occasionally do much better

Payoff of Player A – non-parametrics

Mann-Whitney test comparing Player A’s payoff across groups

U statistic p-value
[Y = 10] < [Y = 30] 13.5 0.260189
[Y = 10] < [Y = 90] 0.0 0.002499
[Y = 30] < [Y = 90] 0.0 0.002461
  • Qualitatively agrees with the nucleolus and disagrees with SV
  • Quantitatively closer to SV in \(Y=90\) treatment (cf. plots)

Exploratory observations

Payoffs by coordination outcomes

2024-10-15T20:22:27.241885 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

  • In the \(Y=90\) treatment, A only achieves significantly higher than ES by excluding one small player
    • Not efficient
    • Why doesn’t the excluded player seem to want the remaining 10 points
  • The outcomes are still not stable
    • excluded player could make a counter-offer

Between matching group variance

2024-10-15T20:21:25.046348 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Testing the axioms

Efficiency

2024-10-15T20:21:17.843641 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

2024-10-15T20:23:06.560801 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Symmetry

2024-10-15T20:23:14.652591 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

2024-10-15T20:23:02.307193 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Dummy player axiom

2024-10-15T20:22:49.553063 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

2024-10-15T20:22:40.479665 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Stability

2024-10-15T20:22:53.156163 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

2024-10-15T20:23:32.348211 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Linearity

2024-10-15T20:23:43.671171 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Linearity

2024-10-15T20:22:17.290000 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

2024-10-15T20:21:41.379788 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Real-time interactions

Time of proposing and accepting final allocation

2024-10-15T20:21:43.040935 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/
  • Most final allocations are proposed within the first minute

Time of accepting final allocation

2024-10-15T20:21:56.487315 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/
  • Most agreement times are well before the end of the bargaining time
  • In the \(Y=90\) treatment, partial agreements come later
  • Otherwise, agreement times are broadly similar across the main treatments

Chat logs

  • Have around 6000 chat messages in total
    • Participants interacted intensively over chat
    • Quality is rather messy (typos, slang, etc.)
  • What to do with it?
    • Find words that are relatively more prevalent depending on the outcome
    • Classify messages into categories (e.g. fairness-related, small talk, etc.)
    • Can also learn things like people’s understanding of the game
  • In general, would probably need much more data for proper NLP analysis

Chat – Stability-based reasoning

2024-10-15T20:22:26.008318 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Y = 10

2024-10-15T20:21:49.680463 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Y = 30

2024-10-15T20:21:37.588683 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Y = 90

Chat – Fairness-based reasoning

2024-10-15T20:22:00.095016 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Equal split

2024-10-15T20:21:27.099849 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

A gets a bit more

2024-10-15T20:21:22.626696 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Rejecting small offers

Chat – Dummy player treatment

2024-10-15T20:21:11.836142 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Altruism

2024-10-15T20:21:26.509802 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Abuse of position

2024-10-15T20:23:16.784936 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Appeal to pity

Chat – Feedback about experiment

2024-10-15T20:22:58.611703 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Having a chat

2024-10-15T20:22:42.587226 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Payouts

2024-10-15T20:22:37.403573 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Bargaining time

Conclusions

What we did

  • Free-form bargaining is important to understand
    • More realistic than strictly controlled designs
    • NCGT have trouble describing it → check if CGT is predictive
  • Players could bargain for 5 minutes without any binding actions
    • An interface for proposing and accepting allocations
    • Group-level unrestricted chat
  • Vary the necessity of including both small players between treatments

Main takeaways

  • Lots of equal splits in all treatments
  • Nucleolus gives qualitatively correct predictions
    • People might think in terms of stability and profitable deviations
  • Both fail quantitatively, especially when the big player has lots of bargaining power
    • SV seems to be closer in the \(Y=90\) case for non-equal splits
  • People’s stated preferences (axiom survey) and actions disagree

Future research

  • Chat logs are useful even in lieu of proper analysis
    • Going through them reveals a lot about players’ understanding of the game
  • We have a rich dataset (real time actions)
    • Good for exploratory analysis and to inform study design in the future
    • Not sure what aspects to look into more deeply – ideas?
  • Differences between matching groups
    • Maybe just player heterogeneity
    • Or norms / reference values determined in the first couple of rounds?

Thank you

References

Chessa, Michela, Nobuyuki Hanaki, Aymeric Lardon, and Takashi Yamada. 2023. “An Experiment on Demand Commitment Bargaining.” Dynamic Games and Applications 13 (2): 589–609.
De Clippel, Geoffroy, and Kareen Rozen. 2022. “Fairness Through the Lens of Cooperative Game Theory: An Experimental Approach.” American Economic Journal: Microeconomics 14 (3): 810–36.
Galeotti, Fabio, Maria Montero, and Anders Poulsen. 2018. Efficiency Versus Equality in Bargaining.” Journal of the European Economic Association 17 (6): 1941–70.
Hart, Sergiu, and Andreu Mas-Colell. 1996. “Bargaining and Value.” Econometrica, 357–80.
Hossain, T., E. Lyons, and A. Siow. 2020. Fairness considerations in joint venture formation.” Experimental Economics 23 (August): 632–67.
Kalisch, G., John Willard Milnor, John F. Nash, and E. D. Nering. 1952. Some Experimental n-Person Games. RAND Corporation.
Leopold-Wildburger, Ulrike. 1992. “Payoff Divisions on Coalition Formation in a Three-Person Characteristic Function Experiment.” Journal of Economic Behavior & Organization 17 (1): 183–93. https://doi.org/https://doi.org/10.1016/0167-2681(92)90086-Q.
Luhan, W. J., O. Poulsen, and M. W. M. Roos. 2019. “Money or Morality: Fairness Ideals in Unstructured Bargaining.” Social Choice and Welfare 53: 655–75. https://doi.org/https://doi.org/10.1007/s00355-019-01206-5.
Maschler, Michael. 1965. “Playing an n-Person Game, an Experiment.”
Michener, H., and Kathryn Potter. 1981. “Generalizability of Tests in n-Person Sidepayment Games.” Journal of Conflict Resolution, 733–49.
Michener, H., Melvin M. Sakurai, Kenneth Yuen, and Thomas J. Kasen. 1979. “A Competitive Test of the M1 (i) and M1 (Im) Bargaining Sets.” The Journal of Conflict Resolution 23 (1): 102–19.
Mitsutsune, Masanori, and Takanori Adachi. 2014. “Estimating Noncooperative and Cooperative Models of Bargaining: An Empirical Comparison.” Empirical Economics 47: 669–93.
Montero, Maria, Martin Sefton, and Ping Zhang. 2008. “Enlargement and the Balance of Power: An Experimental Study.” Social Choice and Welfare 30 (1): 69–87.
Murnighan, J., and A. Roth. 1977. “The Effects of Communication and Information Availability in an Experimental Study of a Three-Person Game.” Management Science 23 (12): 1336–48.
———. 1978. “Large Group Bargaining in a Characteristic Function Game.” Journal of Conflict Resolution 22: 299--317.
Navarro, Noemí, and Róbert F. Veszteg. 2020. “On the Empirical Validity of Axioms in Unstructured Bargaining.” Games and Economic Behavior 121: 117–45.
Nydegger, R. V., and G. Owen. 1874. “Two-Person Bargaining: An Experimental Test of the Nash Axioms.” International Journal of Game Theory 3: 239–49. https://doi.org/https://doi.org/10.1007/BF01766877.
Rapoport, Amnon, and James P Kahan. 1976. “When Three Is Not Always Two Against One: Coalitions in Experimental Three-Person Cooperative Games.” Journal of Experimental Social Psychology 12 (3): 253–73. https://doi.org/https://doi.org/10.1016/0022-1031(76)90056-1.
Schwaninger, Manuel. 2022. “Sharing with the Powerless Third: Other-Regarding Preferences in Dynamic Bargaining.” Journal of Economic Behavior & Organization 197: 341–55. https://doi.org/https://doi.org/10.1016/j.jebo.2022.03.002.
Shinoda, T., and Y. Funaki. 2022. “The Core and the Equal Division Core in a Three-Person Unstructured Bargaining Experiment: The Weakest Coalition Is Ignored.” https://ssrn.com/abstract=4291591.
Tremewan, James, and Christoph Vanberg. 2016. “The Dynamics of Coalition Formation–a Multilateral Bargaining Experiment with Free Timing of Moves.” Journal of Economic Behavior & Organization 130: 33–46.

Appendix

Bargaining interface

Survey (axioms)

Balance (age)

2024-10-15T20:23:13.299171 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Balance (gender)

2024-10-15T20:22:38.678090 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Balance (degree)

2024-10-15T20:22:41.917953 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Balance (study fields)

2024-10-15T20:23:11.591785 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Balance (nationality)

2024-10-15T20:23:20.472364 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/

Balance (difficulty rating)

2024-10-15T20:23:34.217226 image/svg+xml Matplotlib v3.8.4, https://matplotlib.org/